For a simple complex Lie algebra $\mathfrak g$, fixing a principal $\mathfrak{sl}_2$-triple and highest weight vectors induces a basis of $\mathfrak g$ as vector space. For $\mathfrak{sl}_n({\mathbb C})$, we describe how to compute the Lie bracket in this basis using transvectants. This generalizes a well-known rule for $\mathfrak{sl}_2$ using Poisson brackets and degree 2 monomials in two variables. Our proof method uses a graphical calculus for classical invariant theory. Other Lie algebra types are discussed.
1
Department of Mathematics, University of Virginia, Charlottesville, U.S.A.
2
Institut Camille Jordan, Université Lyon 1, Villeurbanne, France
Abdelmalek Abdesselam; Alexander Thomas. Structure Constants for Simple Lie Algebras from a Principal sl2-Triple. Journal of Lie Theory, Tome 34 (2024) no. 4, pp. 829-862. http://geodesic.mathdoc.fr/item/JOLT_2024_34_4_a4/
@article{JOLT_2024_34_4_a4,
author = {Abdelmalek Abdesselam and Alexander Thomas},
title = {Structure {Constants} for {Simple} {Lie} {Algebras} from a {Principal} {sl\protect\textsubscript{2}-Triple}},
journal = {Journal of Lie Theory},
pages = {829--862},
year = {2024},
volume = {34},
number = {4},
url = {http://geodesic.mathdoc.fr/item/JOLT_2024_34_4_a4/}
}
TY - JOUR
AU - Abdelmalek Abdesselam
AU - Alexander Thomas
TI - Structure Constants for Simple Lie Algebras from a Principal sl2-Triple
JO - Journal of Lie Theory
PY - 2024
SP - 829
EP - 862
VL - 34
IS - 4
UR - http://geodesic.mathdoc.fr/item/JOLT_2024_34_4_a4/
ID - JOLT_2024_34_4_a4
ER -
%0 Journal Article
%A Abdelmalek Abdesselam
%A Alexander Thomas
%T Structure Constants for Simple Lie Algebras from a Principal sl2-Triple
%J Journal of Lie Theory
%D 2024
%P 829-862
%V 34
%N 4
%U http://geodesic.mathdoc.fr/item/JOLT_2024_34_4_a4/
%F JOLT_2024_34_4_a4
